Question

The area bounded by the line $y=2x-2$ , $y=-x$ and $x$-axis is given by

• A. $\frac{9}{2}$ sq. units
• B. $\frac{43}{6}$ sq. units
• C. $\frac{35}{6}$ sq. units
• D. none of these

Answer 1

Answer: (D)

We have, $y=2x-2\dots \left(i\right)$
$y=-x\left(ii\right)$
Solving $(i)$ and $(ii)$, we get $x=\frac{2}{3}$, $y=\frac{-2}{3}$

Required area = area of shaded region
$A=\left|\underset{0}{\overset{2/3}{\int }}\left(-x\right)dx\right|+\left|\underset{2/3}{\overset{1}{\int }}\left(2x-2\right)dx\right|$
$=\left|{\left[\frac{-x^2}{2}\right]}_0^{2/3}\right|+\left|{\left[x^2-2x\right]}_{2/3}^1\right|=\frac{4}{18}+\frac{1}{9}=\frac{1}{3}$ sq. units